The Stacks project

Lemma 36.33.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ such that the cohomology sheaves $H^ i(K)$ have countable sets of sections over affine opens. Then for any quasi-compact open $U \subset X$ and any perfect object $E$ in $D(\mathcal{O}_ X)$ the sets

\[ H^ i(U, K \otimes ^\mathbf {L} E),\quad \mathop{\mathrm{Ext}}\nolimits ^ i(E|_ U, K|_ U) \]

are countable.

Proof. Using Cohomology, Lemma 20.47.5 we see that it suffices to prove the result for the groups $H^ i(U, K \otimes ^\mathbf {L} E)$. We will use the induction principle to prove the lemma, see Cohomology of Schemes, Lemma 30.4.1.

First we show that it holds when $U = \mathop{\mathrm{Spec}}(A)$ is affine. Namely, we can represent $K$ by a complex of $A$-modules $K^\bullet $ and $E$ by a finite complex of finite projective $A$-modules $P^\bullet $. See Lemmas 36.3.5 and 36.10.7 and our definition of perfect complexes of $A$-modules (More on Algebra, Definition 15.73.1). Then $(E \otimes ^\mathbf {L} K)|_ U$ is represented by the total complex associated to the double complex $P^\bullet \otimes _ A K^\bullet $ (Lemma 36.3.9). Using induction on the length of the complex $P^\bullet $ (or using a suitable spectral sequence) we see that it suffices to show that $H^ i(P^ a \otimes _ A K^\bullet )$ is countable for each $a$. Since $P^ a$ is a direct summand of $A^{\oplus n}$ for some $n$ this follows from the assumption that the cohomology group $H^ i(K^\bullet )$ is countable.

To finish the proof it suffices to show: if $U = V \cup W$ and the result holds for $V$, $W$, and $V \cap W$, then the result holds for $U$. This is an immediate consquence of the Mayer-Vietoris sequence, see Cohomology, Lemma 20.33.4. $\square$

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